Grade 12 · Algebra & functions

Special limits practice

Special limits is a grade 12 math skill: evaluate special limits, including limits at infinity and limits of indeterminate forms that define key constants and derivatives. Below are 3 practice questions with answers and step-by-step explanations, drawn from the 5 special limits problems our math games drill.

5 questions in the bank

Sample questions

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Question 1easy

Evaluate: $\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x}$ .

This is a standard memorized limit: $\displaystyle\lim_{x \to 0} \dfrac{e^x - 1}{x} = 1$ .

Question 2easy

Evaluate: $\displaystyle\lim_{x \to 0} \dfrac{\tan x}{x}$ .

$\dfrac{\tan x}{x} = \dfrac{\sin x}{x} \cdot \dfrac{1}{\cos x} \to 1 \cdot 1 = 1$ .

Question 3easy

Evaluate: $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos x}{x^2}$ .

Using the approximation $1 - \cos x \approx \dfrac{x^2}{2}$ near $0$ : $\dfrac{1 - \cos x}{x^2} \to \dfrac{1}{2}$ .

More Grade 12 skills:

Polar and rectangular conversion Standard antiderivatives Maclaurin series Standard derivatives Inverse trig domain and range Trig identities Degree-radian conversion Convergence tests All Grade 12 practice →

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